The last few months have seen the discovery of a second Martian Trojan(1998 VF31), as well as two further possible candidates (1998 QH56 and 1998SD4). Together with the previously discovered Martian satellite 5261 Eureka,these are the only known possible solar system Trojan asteroids not associatedwith Jupiter. Here, maps of the locations of the stable Trojan trajectoriesof Mars are presented. These are constructed by integrating an ensemble of in-plane and inclined orbits in the vicinity of the Martian Lagrange pointsfor between 25 million and 60 million years. The survivors occupy a band of inclinations between 15

◦

and 40

◦

and longitudes between 240

◦

and 330

◦

at the

L

5

Lagrange point. Around the

L

4

point, stable Trojans inhabit two bandsof inclinations (15

◦

< i <

30

◦

and 32

◦

< i <

40

◦

) with longitudes restrictedbetween 25

◦

and 120

◦

. Both 5261 Eureka and 1998 VF31 lie deep within oneof the stable zones, which suggests they may be of primordial origin. AroundMars, the number of such undiscovered primordial objects with sizes greaterthan 1 km may be as high as

∼

50. The two candidates 1998 QH56 and 1998SD4 are not presently on Trojan orbits and will enter the sphere of inﬂuence of Mars within half a million years.

are stable in the restricted three body problem (e.g.,Danby 1988). However, the long-term survival of Trojans around the Lagrange points of the planets in the presence of perturbations from the remainder of the Solar System is adiﬃcult and still unsolved problem (e.g.,´Erdi 1997). Jovian Trojan asteroids have beenknown since the early years of this century, while a number of Saturnian moons (e.g., Dioneand Helene, Tethys and Calypso, Tethys and Telesto) also form Trojan conﬁgurations withtheir parent planet. However, whether there exist Trojan-like bodies associated with theother planets has been the matter of both observational activity (e.g., Tombaugh 1961,Kowal 1971) and intense theoretical speculation (e.g., Weissman & Wetherill 1974, Mikkola& Innanen 1990, 1992). The answer to this problem came in 1990, with the discovery of 5261 Eureka, the ﬁrst Trojan around Mars (see Mikkola et al. 1994 for details). The lastfew months of 1998 have seen further remarkable progress with the discovery of one certainMartian Trojan, namely 1998 VF31, as well as two further candidates, namely 1998 SD4and 1998 QH56 (see the

Minor Planet Electronic Circulars 1998-W04, 1998-R02, 1998-S20

and the

Minor Planet Circular 33085

). The suggestion that 1998 QH56 and 1998 VF31might be Martian Trojans was ﬁrst made by G.V. Williams.These recent discoveries raise very directly the following questions. Are there any moreMartian Trojans? If so, where should the observational eﬀort be concentrated? Of course,the ﬁrst question can only be answered at the telescope, but the resolution of the secondquestion is provided in this

Letter

. By integrating numerically an ensemble of inclinedand in-plane orbits in the vicinity of the Martian Lagrange points for 25 and 60 millionyears respectively, the stable r´egimes are mapped out. On re-simulating and samplingthe ensemble of stable orbits, the probability density of Martian Trojans as a function of longitude and inclination can be readily obtained. If a comparatively puny body such as